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In this paper we consider the local X-ray transform for general flows. Weextend the results on the local and global invertibility of the geodesic raytransform proved by Uhlmann and Vasy \cite{UV} to the X-ray transform for ageneral flow. The key improvement is that our argument for the ellipticity ofthe conjugated operator $A_F$ (which is defined in Section 2) can be applied toflows other than the geodesic flow.

The inverse problem for the local ray transform